Alice in Quantum Land.pdf

Alice
in
Quantumland
AN ALLEGORY OF QUANTUM PHYSICS

Alice
in
Quantumland
Robert Gilmore
COPERNICUS
AN IMPRINT OF SPRINGER-VERLAG
© 1995 Springer-Verlag New York, Inc. All rights reserved. No part of this publication may be
reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic,
mechanical, photocopying, recording, or otherwise, without the prior written permission of the
publisher. Published in the United States by Copernicus Books, an imprint of Springer-Verlag New
York, LLC by arrangement with Birkhauser Boston. A member of Springer Science+Business Media
Copernicus Books 37 East 7th Street New York, NY 10003
www.copernicusbooks.com Library of Congress Cataloging-in-Publication Data Gilmore, Robert,
1938Alice in Quantumland : an allegory of quantum physics / Robert Gilmore p. cm. Includes
bibliographical references and index. ISBN 0-387-91495-1 (acid-free paper) 1. Quantum theory-
Fiction. I. Title PS3557.14595A45 1995 813 x.54- dc20 95-10163
Manufactured in the United States of America. Printed on acid-free paper. 19 18 17 16 15 14 13 12
11 ISBN 0-387-91495-1 SPIN 10992432

Preface
n the first half of the twentieth century, our understanding of the Universe was turned upside
down. The old classical theories of physics were replaced by a new way of looking at the world-
quantum mechanics. This is in many ways at variance with the ideas of the older Newtonian
mechanics; indeed, in many ways it is at variance with our common sense. Nevertheless, the strangest
thing about these theories is their extraordinary success in predicting the observed behavior of
physical systems. However nonsensical quantum mechanics may at times appear to us, that seems to
be the way that Nature wants it-and so we have to play along.
This book is an allegory of quantum physics, in the dictionary sense of "a narrative describing
one subject under the guise of another." The way that things behave in quantum mechanics seems very
odd to our normal way of thinking and is made more acceptable when we consider analogies to
situations with which we are familiar, even though the analogies may be inexact. Such analogies can
never be very true to reality as quantum processes are really quite different from our normal
experience.
An allegory is an extended analogy, or series of analogies. As such, this book follows more in
the footsteps of Pilgrim's Progress or Gulliver's Travels than of Alice in Wonderland. "Alice"
appears the more suitable model, however, when we examine the world that we inhabit.
The Quantumland in which Alice travels is rather like a theme park in which Alice is sometimes
an observer, while sometimes she behaves as a sort of particle with varying electric charge. This
Quantumland shows the essential features of the quantum world: the world that we all inhabit.
Much of the story is pure fiction and the characters are imaginary, although the "real-world"
notes described below are true. Throughout the narrative you will find many statements that are
obviously nonsensical and quite at variance with common sense. For the most part these are true.
Neils Bohr, the father figure of quantum mechanics in its early days, is said to have remarked that
anyone who did not feel dizzy when thinking about quantum theory had not understood it.
Seriously, Though . . .
The description of the world that is given by quantum mechanics is undoubtedly interesting and
remarkable, but are we seriously expected to believe that it is true? Amazingly, we find that we must.
To underline this assertion, throughout this book you will find brief notes which emphasize the
importance of quantum mechanics in the real world. The notes look like this:

There are also some longer, end-of-chapter, notes. These amplify some of the trickier points in
the text and are denoted thus:
See end-of-chapter note 1
Much of the way that quantum theory describes the world may seem at first sight to be nonsense-
and possibly it may seem so at the second, third, and twenty-fifth sight as well. It is, however, the
only game in town. The old classical mechanics of Newton and his followers is unable to give any
sort of explanation for atoms and other small systems. Quantum mechanics agrees very well with
observation. The calculations are often difficult and tedious, but where they have been made, they
have agreed perfectly with what has really been seen.
It is impossible to stress too strongly the remarkable practical success of quantum mechanics.
Although the outcome of one measurement may be random and unpredictable, the predictions of
quantum theory agree consistently with the average results obtained from many measurements. Any
large-scale observation will involve very many atoms and thus very many observations on the atomic
scale. We again find that quantum mechanics is successful, in that it automatically agrees with the
results of classical mechanics for large objects. The converse is not true.
Quantum theory was developed to explain observations made on atoms. Since its conception, it
has successfully been applied to atomic nuclei, to the strongly interacting particles which derive from
the nucleus, and to the behavior of the quarks of which these are composed. The application of the
theory has been extended over a factor of some hundred thousand million. The systems considered
have both decreased in size and increased in energy by this factor. This is a long way to extrapolate a
theory from its original conception, but so far quantum mechanics appears to be quite able to deal
with these extreme systems.
Insofar as it has been investigated, quantum mechanics appears to be of universal applicability.
On a large scale, the predictions of quantum theory lose their random aspect and agree with those of
classical mechanics, which works very well for large objects. On a small scale, however, the
predictions of quantum theory are consistently borne out by experiment. Even those predictions,
which seem to imply a nonsensical picture of the world, are supported by experimental evidence.

Intriguingly, as is discussed in Chapter 4, quantum mechanics would appear to be in the strange
position of agreeing with all observations made, while disputing that any observations can actually be
made at all. It seems that the world is stranger than we imagine and perhaps stranger than we can
imagine.
For the present, however, let us accompany Alice as she begins her journey into Quantumland.
Robert Gilmore

Contents
Into Quantumland
The Heisenberg Bank
The Mechanic's Institute
The Copenhagen School
The Fermi-Bose Academy
Virtual Reality
Atoms in the Void
Castle Rutherford
The Particle MASSquerade
The Experimental Physics Phun Phair

lice was bored. All her friends were on holiday or visiting relations and it was raining,
so that she was marooned indoors watching television. So far that afternoon she had watched part five
of a series on introductory Esperanto, a program on gardening, and a paid political broadcast. Alice
was really bored.
She looked down at the book lying on the floor beside her chair. It was a copy of Alice in
Wonderland, which she had been reading earlier and had dropped there when she finished it. "I do
not know why there cannot be more cartoons and interesting programs on the television," she
wondered idly to herself. "I wish I could be like that other Alice. She was feeling bored and then she
found her way to a land full of interesting creatures and strange happenings. If I could shrink down
somehow and float through the television screen perhaps I might find all sorts of fascinating things."
She stared in frustration at the screen, which at that moment carried a picture of the Prime
Minister telling her how, all things considered, everything was really far better than it had been three
years ago, even if it didn't always seem that way. As she watched she was mildly surprised to see the
picture of the Prime Minister's face slowly break apart into a mist of bright dancing speckles which
all seemed to be rushing inward, as if they were beckoning her. "Why," said Alice, "I do believe that
they want me to follow them in!" She leapt to her feet and started toward the television, but tripped on
the book which she had discarded so untidily on the floor and fell headlong.

As she fell forward she was amazed to see the screen grow enormously, and she found herself in
among the swirling speckles, rushing with them down into the picture. "I cannot see anything with
these dots swirling all around me," thought Alice. "It is just like being lost in a snowstorm; why I
cannot even see my feet. I wish I could see just a little. I could be anywhere."
At that moment Alice felt her feet strike something solid and she found herself standing on a
hard, flat surface. All around her the swirling dots were fading away and she found that she was
surrounded by a number of vague shapes.
She looked more closely at the one nearest to her and observed a small figure, coming roughly
up to her waist. It was exceedingly difficult to make out, as all the time it kept hopping rapidly to-and-
fro, moving so fast that it was very difficult to see at all clearly. The figure seemed to be carrying
some sort of stick, or possibly a rolled umbrella, which was pointing straight up in the air. "Hello,"
Alice introduced herself politely. "I am Alice. May I ask who you are?"
"I am an electron," said the figure. "I am a spin-up electron. You can readily tell me apart from
my friend there who is a spin-down electron, so, of course, she is quite different." Under his breath he
added something which sounded rather like "Vive la difference!" As far as Alice could see, the other
electron looked very much the same, except that her umbrella, or whatever it was, was pointing down
toward the ground. It was very difficult to tell for sure, as this figure also was jigging to-and-fro as
rapidly as the first.

"Oh please," said Alice to her first acquaintance. "Would you be good enough to stand still for a

moment, as I really cannot see you at all clearly?"
"I am good enough," said the electron, "but I am afraid there is not room enough. However I will
try." So saying he slowed his rate of jiggling. But as he moved more slowly, he began to expand
sideways and become more and more diffuse. Now, although he was no longer moving at all quickly,
he looked so fuzzy and quite out of focus that Alice could no more see what he looked like than she
had been able to before. "That is the best I can do," he panted. "I am afraid that the more slowly I
move, the more spread out I become. That is the way things are here in Quantumland: The smaller the
space you occupy, the faster you have to move. It is one of the rules, and there is nothing I can do
about it."
"There isn't really room to slow down here," continued Alice's companion as he began once
more to leap rapidly around. "The platform is becoming so crowded that I have to be more compact."
Sure enough, the space in which Alice stood had now become very crowded indeed, being closely
packed with the small figures, each dancing feverishly to-and-fro.
"What strange beings," thought Alice. "I do not think I shall ever be able to see quite what they
look like if they will not stand still for a minute, and there does not seem to be much chance of that."
Since it did not look as if she could get them to slow down she tried another topic. "Would you tell
me please what sort of platform we are on?" she asked.
"Why a railway platform, of course," replied one of the electrons cheerfully (it was very hard
for Alice to say which had spoken; they really did all look very much the same). "We are going to
take the wave train to the screen you see. You will change there to the photon express I expect, if you
want to go any farther."
"Do you mean the television screen?" asked Alice.
"Why of course I do," cried one of the electrons. Alice could have sworn that it was not the same
one which had just spoken, but it was very difficult to be certain. "Come on! The train is here and we
have to get on."
Sure enough, Alice could see a line of small compartments drawn up at the platform. They were

very small. Some were empty, some had one electron in, and some two. All of the empty
compartments were filling rapidly-in fact there did not seem to be any left-but Alice noticed that not
one of the compartments held more than two electrons. As they passed by any of these compartments,
the two occupants would cry out "No room! No room!"
"Surely you could squeeze more than two into a compartment, seeing as the train is so
crowded?" Alice asked her companion.
"Oh no! Never more than two electrons together, that is the rule."
"I suppose we shall have to get into different compartments then," declared Alice regretfully, but
the electron reassured her.
"There's no problem there for you, no problem at all! You can get into any compartment that you
want, of course."
"I am sure that I do not see why that should be," Alice replied. "If a compartment is too full to
hold you, then it must surely be too full for me as well."
"Not at all! The compartments are only allowed to hold two electrons, so almost all the places
for electrons may be taken up, but you are not an electron! There is not a single other Alice on the
train, so there is plenty of room for an Alice in any of the compartments."
This did not seem to follow so far as Alice could see, but she was afraid that the train would
start to move off before they got seats, so she began looking for an empty space that could take
another electron. "How about this one?" she asked her associate. "Here is a compartment with only
one other electron already in it. Can you get in here?"
"Certainly not!" he snapped, sounding quite horrified. "That is another spin-up electron. I cannot
share a compartment with another spin-up electron. What a suggestion! It is quite against my
principle."
"Don't you mean against your principles?" Alice asked him.
"I mean what I say, against my principle, or rather Pauli's principle. It forbids any two of us
electrons from doing exactly the same thing, which includes being in the same space and having the
same spin," he responded crossly.

Alice did not really know why she had upset him, but she looked around hastily to find another
compartment which might suit him better. She managed to find one that held a single electron who
was of the spin-down variety, and Alice's companion leaped into this one readily enough. Alice was
surprised to find that although the tiny compartment now seemed full there was somehow enough
room for her to fit in quite easily.
No sooner were they settled in than the train moved off. The journey was uneventful and the
scenery not very interesting, so Alice was rather glad when the train began to slow down. "This must
be the screen, I suppose," thought Alice. "I wonder what will happen here."
As they alighted at the screen there was an enormous bustle everywhere. "Whatever is going
on?" Alice wondered aloud. "Why does everyone seem to be so excited?" Her questions were
answered by an announcement which appeared to come from the air all around her.
"The screen phosphor is presently being excited by the incoming electrons, and we shall be
having photon emission soon. Stand by for the departure of the photon express." Alice looked around
to see if she could see the express arriving, when there was a rush of bright shining shapes across the
platform. Alice was caught up in the middle of the crowd and carried along with them as they all
crowded into one compartment. "Well, they do not seem to be worried by any principle, Pauli or
otherwise," thought Alice as they crowded in around her. "These ones are certainly not worried about
all being in the same place. I suppose the express is going to start soon. I wonder where ...
"... we shall end up," she concluded as she stepped out onto the platform. "My, that was certainly
a fast journey. Why, it seemed to take no time at all." (Alice was quite right about this. The journey
did indeed take no time at all, as time is effectively frozen for anything which is traveling at the
velocity of light.) Once again she found herself surrounded by a crowd of electrons, all rushing away
from the platform.
"Come along!" one of them cried to her as it rushed off. "We must get out of the station now if
we are to get anywhere."
"Excuse me," Alice asked it tentatively, "are you the same electron that I was talking to before?"

"Yes I am," answered the electron as it darted off down a side passage. Alice was swept along
by the crowd of electrons and carried through the main gateway from the platform.
"I declare, this is really too irksome," said Alice. "Now I have lost the only person I know at all
in this strange place and have no one to explain what is happening."
"Don't worry Alice," said a voice from about knee level. "I will show you where to go." It was
one of the electrons.
"How do you know my name?" asked Alice in surprise.
"That's simple. I am the same electron that spoke to you before."
"You cannot be!" exclaimed Alice. "I saw that electron go off in a different direction. Perhaps he
was not the same one I was talking to before?"
"Certainly he was."
"Then you cannot be the same one," said Alice reasonably. "You cannot both be the same one

you know."
"Oh yes we can!" replied the electron. "He is the same. I am the same. We are all the same, you
know, exactly the same!"
"That is ridiculous," argued Alice. "You are here beside me, while he has run off somewhere
over there, so you cannot both be the same person. One of you must be different."
"Not at all," cried the electron, jumping up and down even faster in its excitement. "We are all
identical; there is no way whatsoever that you can tell us apart, so you see that he must be the same
and I am the same too."
At that point the crowd of electrons which surrounded Alice all began to cry out, "I am the
same," "I am the same too," "I am just the same as you are," "I am too, just the same as you." The
tumult was dreadful, and Alice closed her eyes and put her hands over her ears until the noise died
down again.
When it was quiet again Alice opened her eyes and lowered her hands. She found there was no
sign of the crowd of electrons which had been clustering around her and that she was walking out of
the station entrance all alone. Looking around she found herself in a street which at first sight seemed
quite normal. She turned left and began to walk along the sidewalk.
Before she had gone very far she came across a figure standing dejectedly in front of a doorway
and searching though his pockets. The figure was short and very pale. His face was difficult to make
out distinctly, as was the case for everyone Alice had met recently, but he did look, Alice thought,
rather like a rabbit. "Oh dear, oh dear, I am late and I cannot find my keys anywhere. I must get inside
straightaway!" So saying he stepped back a few paces and then ran quickly toward the door.
He ran so very fast that Alice was not able to see him in any one position, but saw instead a
string of afterimages which showed him at all the different positions he passed through along his path.
These extended from his starting point to the door, but there, instead of stopping as Alice would have
expected, they continued on into the door, getting smaller and smaller until they were too small to be
seen. Alice had scarcely had time to register this strange series of images when he rebounded
backward just as rapidly, once again leaving a series of images. This time they ended abruptly with
the unfortunate person sprawled on his back in the gutter. Apparently in no way discouraged, he
picked himself up and raced toward the door again. Again there was the series of afterimages,
shrinking away into the door, and again he bounced off and ended up on his back.
As Alice hurried toward him he repeated this action several more times, throwing himself at the
door and then falling back again. "Stop, stop," cried Alice. "You must not do that; you will surely hurt
yourself."
The person stopped his running and looked at Alice. "Why, hello my dear. I must do this I'm
afraid. I am locked out and I must get inside quickly, so I have no choice but to try and tunnel through
the barrier."

Alice looked at the door, which was very large and solid. "I do not think you have much chance
of getting through that by running at it," she said. "Are you trying to break it down?"
"Oh no, certainly not! I do not want to destroy my beautiful door. I just want to tunnel through it. I
am afraid that what you say is true, though. The probability of my managing to get through is indeed
not very high at all, but I have to try." As he said this he charged at the door again. Alice gave him up
as a bad job and walked off, just as he came staggering back once more.
After she had walked a few paces, Alice could not resist looking back to see if by any chance he
had abandoned his efforts, and she saw again the series of images rushing toward the door and
shrinking down when they got to it. She waited for the rebound. Previously this had followed
immediately after, but this time it did not happen. The door stood there looking solid and rather
deserted, but there was no sign of her acquaintance. After a few seconds had passed with nothing
happening, Alice heard a rattling of bolts and chains from behind the door and then it swung open.
Her vanished companion looked out and waved to her. "I was really in luck!" he called. The
probability of penetrating a barrier this thick is very small indeed, and I was amazingly fortunate to
get through so quickly." He closed the door with a solid thump and that seemed to end the encounter,
so Alice walked on up the street.
A little farther along she came to an empty plot by the side of the road, where a group of builders
was clustered around a pile of bricks. Alice assumed they were builders, as they were unloading
more bricks from a small cart. "Well at least these people seem to be behaving in a sensible manner,"
she thought to herself. Just then another group came running around a corner carrying what looked like
a very large rolled-up carpet and proceeded to spread it out on the site. When it was unrolled Alice
could see that it was some sort of building plan. It did seem to be rather a large plan since it covered
most of the available space. "Why, I do believe it must be exactly the same size as the building they
are going to put up," said Alice, "but how will they manage to build anything if the plan is already
taking up all the room?"
The builders had finished easing the plan into position and had retreated to the pile of bricks.
They all picked up bricks and began throwing them at the plan, apparently quite at random. All was
confusionsome fell in one place, some in another-and Alice could see no purpose in it at all. "What
are you doing?" she asked a person who was standing to one side. He appeared to be doing nothing,
and she assumed him to be the foreman. "You are just making untidy piles of bricks. Aren't you
supposed to be putting up a building?"

"Ah, sure, and we are, me darling," answered the foremen. "It's true so it is that the random
fluctuations are still large enough to hide the pattern, but since we have laid down the probability
distribution for the result we are after needing, we'll be getting there, never fear."
Alice felt that this display of optimism was not very convincing, but she kept her peace and
watched as the shower of bricks continued to descend onto the site. Gradually, to her amazement, she
noted that more bricks were falling in some regions than in others, and she could begin to make out
the patterns of walls and doorways. She watched in fascination as the recognizable shape of rooms
began to appear out of the initial chaos. "Why, that is amazing," she cried. "How have you managed to
do that?"
"Well now, haven't I already told you," smiled the foreman. "You watched us lay down the
probability distribution before we began. This specifies where there should be bricks and where
there should be none. We must do this before we start bricklaying as we cannot tell where each brick
will go when we throw it, you know," he continued.
"I do not see why!" Alice interrupted him. "I am used to seeing bricks being laid in place one
after another in neat lines."

"Well now, that is not the Quantum way. Here we cannot control where each individual brick
goes, only the probability that it will go one place or another. This means that when you have only a
few bricks, they can go almost anywhere and seem to have no sort of pattern at all. As the number
becomes large, however, you find that there are bricks only where there is some probability that they
should be there, and where the probability is higher, there you get more bricks. When you have large
numbers of bricks involved it all works out very nicely in the end, so it does."
Alice found this all very peculiar, although the foreman spoke so definitely that it sounded as if it
might make some sort of strange sense. She did not ask any more questions at this time, as his answers
only made her feel more confused than ever, so she thanked him for his information and went on down
the road.
Before long she came to a window in which was displayed a large notice:
"That all sounds very enticing I am sure, but I have no idea what it is talking about, and if I were
to ask someone I am sure the answer would leave me even worse off than I am now," exclaimed

Alice in desperation. "I have not really understood anything that I have seen so far. I wish I could find
someone to give me a good explanation of what is going on around me."
She had not realized that she had spoken aloud, until she was answered by a passerby. "If you
want to understand Quantumland you will need to find someone to explain to you about quantum
mechanics. For that you ought to go to the Mechanics Institute," she was advised.
"Oh, will they be able to help me understand what is happening here?" cried Alice in delight.
"Will they be able to explain all the things I have seen, such as that notice in the window there, and to
tell me what these eV' are?"
"I should think the Mechanics will be able to give you an explanation for most of it," answered
her informant, "but as 'eV' are units of energy you had probably best start by asking about them at the
Heisenberg Bank, particularly as it is just across the road there."
Alice looked across where he was pointing and saw a large building with a very formal

frontage, obviously designed to impress. It had a tall portico with stone pillars and over the top, in
large letters, was carved the name THE HEISENBERG BANK. Alice crossed the road, climbed the
long flight of stone steps which led up to the lofty doorway, and passed through.

hen Alice stepped through the doorway she found herself in a large pillared hall
with marble walls. It looked much like other banks that she had seen only more intensely so, as it
were. There was a line of cashiers' windows along the far wall, and the vast floor area was divided
up by portable tape barriers so that the customers would be guided into regular lines as they waited to
be served. At the moment however the place appeared to be quite empty of any customers at all.
Apart from the cashiers behind their counters and a bank guard standing by the door, Alice did not see
anyone.
As she had been advised to ask for information at the Bank, she began to walk purposefully
toward the distant line of windows. "Now just you wait a minute!" called the guard by the door.
"Where do you think you are going, young miss? Can't you see that there is a line?"
"I am sorry," replied Alice, "but actually I can't see a line. There are no people here."
"There certainly are, and a lot of them!" answered the guard emphatically. "We seem to have
quite a rush of 'no people' today. Usually though we refer to them as virtual. I have seldom seen quite
so many virtual particles waiting to get their energy loans."

Alice had a by-now-familiar feeling that things were not going to become all that much clearer
very quickly. She looked over at the windows and saw that, although the room still appeared to be
quite empty, the cashiers were all very busy. As she watched, she saw bright figures appear, one at a
time, in front of one till or another and then rush quickly from the Bank. At one till she saw a pair of
figures materialize together in front of a grill. One she recognized as an electron; the other was very
similar, but was a sort of photographic negative of the first, opposite in every way to the electrons she
had seen previously.
"That is a positron, an antielectron," murmured a voice in her ear. Alice looked around and saw
a severe-looking, smartly dressed young woman.
"Who are you?" she asked.
"I am the Bank Manager," replied her companion. "I am in charge of the distribution of energy
loans to all the virtual particles here. Most of them are photons, as you can see, but sometimes we get
pairs of particles and antiparticles who come along together to ask for a loan, like the electron and
positron pair that you were looking at just now."
"Why do they need an energy loan?" asked Alice. "And why can't I see them before they get it?"
"Well you see," replied the Manager, "in order for a particle to exist properly, so that it can be a
free particle and able to move around and be observed normally and so on, it has to have, at the very
least, a certain minimum energy which we call its rest mass energy. These poor virtual particles do
not have even that energy. Most of them have no energy at all, so they do not really exist. Fortunately
for them, they can get a loan of energy here at the Bank and this allows them to exist for a little
while." She pointed to a notice on the wall, which read:
"That is called the Heisenberg relation. It governs all our transactions. The value ħ is called
Planck's constant, the correctly reduced value, of course. The relation gives the rate of exchange for
our energy loans. The quantity ΔE is the amount of energy which is borrowed, and Δt is the period for
which the loan is made you see."
"You mean," said Alice, trying to follow what the Manager was saying, "that it is like an
exchange rate between different types of money, so that the more time there is, the more energy they
can have?"
"Oh no! Quite the reverse! It is the energy and time multiplied together which are constant, so
the greater the amount of energy, the shorter the amount of time they are allowed to keep it. If you
want to see what I mean just look at that exotic particle and antiparticle which have just taken out a

loan at window #7."
Alice looked where she was directed and observed a striking sight. In front of the window was a
pair of figures; one was the opposite of the other, in much the same way as for the electron and
positron that she had seen earlier. This pair, however, were bright, flamboyant figures, taking up so
much space with their presence that they quite obscured the cashier behind them. Alice could not but
be impressed by the extravagance of the two, but as she opened her mouth to make a comment they
grew hazy and then vanished completely.

"That is an illustration of what I was saying," continued the Manager calmly. "That pair took out
an enormous energy loan to support the huge rest mass that they needed for their lifestyle. Because the
loan was so large, the repayment time was very short indeed, so short that they did not even manage
to leave the counter before it had to be repaid. Because such heavy particles cannot get very far
before they have to repay their energy loan, they are known in the trade as short-range particles," she
added.
"Is the relation between time and energy the same for everyone then?" asked Alice, who felt that
she might have discovered something definite at last.
"Yes indeed! The Planck constant is always the same whenever and wherever it applies. It is
what is called a universal constant, which simply means it is always the same everywhere.
"We deal with energy at the Bank here," continued the Manager, "because energy acts as the
currency here in Quantumland. As you would express your currency in pounds or dollars, the unit of
energy that we use most of the time is called the eV. How much energy a particle has determines what
it is able to do; how fast it can go, what state it can get into, how much it will be able to affect other
systems, these all depend on the energy it has.
"Not all particles are completely destitute like the ones that are lining up. Many of them do have
sufficient energy of their own, and in that case they can keep it for as long as they like. Those are the
ones which you may see moving around outside. Any particle which needs to have a mass has to have
energy just to exist at all."
She pointed at another framed notice on the wall, which read:
"If a particle wants to have mass then it must find the energy to support it somehow. If it has any
energy left over then it can use it to do other things. Not that all particles bother with mass," she
added. "There are some free-and-easy, bohemian particles which do not have any rest mass at all.

They are not tied down like most particles who have to provide for their mass, so they can make use
of even small amounts of energy. Photons are a case in point. A photon has no rest mass, so a photon
at rest would not weigh anything at all. Mind you, you do not normally find photons at rest; they are
forever rushing about at the speed of light, as photons are what light is made of you see. Light is not a
smooth continuous stream. It is made up of a lot of quanta, little packets of energy, so that the flow of
light is lumpy. These quanta, or particles, of light are called photons. Practically everything comes in
quanta of some size. This gives quantum physics its name, you know. Look at all those photons
leaving the Bank now. Basically photons are all the same, exactly like one another in the way that
electrons are all the same, but you may notice that many of these photons seem quite different. That is
because they have different amounts of energy. Some of them have very little energy, like those radio
frequency photons going out now."
Alice looked down at a crowd of photons which were rushing past her, flowing around her feet
and on out through the door. As they went, she heard snatches of music, dramatic voices, and
something about "doing lunch on Thursday." "I didn't know that radio waves were made up from
photons," admitted Alice. "Oh yes indeed. They are very long wavelength photons of course, with
low frequency and very little energy. They are very gregarious because if they are to have any
noticeable effect you need a lot of them at once. Friendly little fellows aren't they?" smiled Alice's
companion. "Visible photons now, the ones which make up the light that people use to see by, they
have higher frequency and more energy. One of those can have quite a noticeable effect. The really
affluent ones though, the big spenders, are the X-ray and gamma photons. Each one of those carries a
lot of energy around with it and they can really make their presence felt on their surroundings if they
choose to interact."
"That is certainly very interesting," said Alice, not entirely untruthfully, "but I still feel confused
about the whole idea of energy. Can you tell me what energy actually is?"
"Well now," replied the Manager with satisfaction, "that is a very sensible question to ask.
Unfortunately it is not a very easy one to answer. Come into my office and I will try to give you an
explanation."
The Manager led Alice briskly across the tiled floor of the main hall and through an unobtrusive
but rather forbidding door in one corner. Within was a large modern office. Motioning Alice to sit on
a deep comfortable chair placed in front of the wide desk, the Manager went round and sat in the
chair behind it.
"Well," she began, "energy is a little bit like money in your world and it is not too easy to say
exactly what that is either."
"I should think that was quite easy," responded Alice. "Money is coins, like my pocket money, or
it can be bank notes."
"That is cash, which is certainly one form of money, but money does not have to be in notes and
coins. It can be in a savings account, for example, or in stocks and shares, or even invested in a
building. In much the same way energy can take many forms, which seem quite different from one
another.

"The most obvious form is kinetic energy," said the Manager, as she settled more comfortably
into her chair and her voice took on the complacent tone of someone who is about to give a long
lecture to a captive audience.
"A particle, or any other object for that matter, will have kinetic energy if it is moving about.
Kinetic just means moving, you know. There are other forms of energy as well. There is potential
energy, such as the gravitational energy which a stone has if it is up a hill and so is in a position to
roll down it. You can also have electrical energy, or chemical energy, which is just potential energy
which the electrons have when they are inside atoms. Then, as I have already mentioned, there is the
rest mass energy which many particles must have just to exist, so that they can have some mass. One
form of energy can convert into another, just as you can pay cash into your deposit account. I can
illustrate that for you if you will just look through the round window." She leaned over and pressed a
button on her desk, and a round window on the wall in front of Alice opened up. Through it Alice
could see a fairground roller coaster. As she watched, a carriage climbed to the top of one hump and
paused there momentarily before it rushed down the far side.

"That carriage, as you can see, is not moving at the moment, so it has no kinetic energy, but it is
high up so it has potential energy because of its position. Now as it starts to fall down into the dip it
is losing height, so it loses some of that potential energy. This is converted to kinetic energy, so as it
falls it goes faster and faster." Alice could vaguely hear the happy excited shrieks of the distant
passengers in the carriage as it thundered down the track.
"If the track were very smooth and the wheels ran without friction," the lecturer continued
dispassionately, "then the carriage would come to rest again at exactly the same height." She leaned
over and fiddled with something on her desk again. The distant figures on the roller coaster cried out
in surprise as the next hump in the track suddenly surged up before them to a much greater height.
Their carriage slowed and came to a complete stop before it had reached the top. "How did you do
that?" exclaimed Alice in amazement. "Never underestimate the influence of a bank," muttered her
companion. "Now see what happens."
The carriage began to roll backward down the track, accompanied by more shrieks, still excited
but not quite as happy as last time. It gathered speed until it shot through the lowest point and then
climbed the opposite slope, slowing as it went. It came to rest just at the peak where Alice had first
seen it and then began to slip back down once again.

"This will go on indefinitely now, with the energy of the carriage changing from potential energy
to kinetic energy and hack again, but you get the idea." The Manager pressed another button on her
desk and the window closed on the scene.
"That is the sort of obvious way in which you see energy in ClassicWorld. It will change from
one form to another in a smooth continuous manner. You saw how the carriage got steadily faster as it
rolled steadily down the slope, with no big jumps, and there are no obvious restrictions on the exact
amount of energy which any object might have. Here in Quantumland it is often not like this. In many
situations a particle is only allowed to have one of a restricted set of values and it can only accept or
give up energy in large lumps, which we call quanta. In ClassicWorld all energy payments are made
on the installment plan, with very frequent and very very tiny payments, but here they often have to be
made as a lump sum.
"As you saw, kinetic energy is a dramatic, showoff sort of energy―something which a body has
just because it is moving. The more massive it is the more kinetic energy it has, and the faster it
moves the more kinetic energy it has, but the amount does not depend at all on the direction in which
it is moving, only on the speed. In this respect it is different from another important quantity which
tells us how a particle moves. This is something we call momentum. Momentum is a sort of measure
of a particle's obstinacy. Every particle is determined that it is going to keep on moving in exactly the
same way as it was before, without any change at all. If something is moving fast it takes a lot of force
to slow it down. It also takes a lot of force to make it move in a different direction, even if its speed
does not change. Now a change in direction does not cause a particle to lose any of its precious
kinetic energy, as this depends only on how fast it is traveling, but it still does not want to change
because its momentum would have to be different. Particles are rather conservative that way.

"It is all a question of what we call parameters," continued the Manager enthusiastically. "When
you want to describe a particle, you have to use the right parameters. If you want to say where it is
you must talk about its position and time, for example."
"I would have thought that you would just need to say what its position was," objected Alice.
"That will tell you where it is, surely?"
"No, certainly not. You must give the time as well as the position. If you want to know where
something is now, or where it will be tomorrow, it is no good my only telling you a position if that is
where it was last week. You must know the position and the time, as things tend to move around you
know. Just as if you want to know what a particle is doing you must describe that in terms of its
momentum and energy, in general you need to give both position and time if you want to know where
a particle is."

"Here in Quantumland the parameters tend to be related. If you try to see where something is then
that has an effect on its momentum, how fast it is moving. It is another form of the Heisenberg relation
which I pointed out to you in the Bank."
"Oh!" cried Alice, remembering a previous encounter. "Was that the reason that the electron I
saw earlier could not stand still to let me see him without becoming all fuzzy?"
"Yes, undoubtedly. The uncertainty relations affect all particles that way. They always seem a
bit indefinite, and you can never pin them down too precisely.
"I know what I shall do! I shall get the Uncertain Accountant to explain it to you," exclaimed the
Manager. "His job is to try and balance the accounts, so he has to worry all the time about quantum
fluctuations." She reached out an elegant finger and pressed yet another of the buttons with which her
desk was so well supplied.
There was a short pause, and then one of the doors which were spaced around the room opened
and a figure entered. He looked rather like a picture of Ebenezer Scrooge from an illustrated copy of
A Christmas Carol, except that he had a rather bemused expression on his face and an uncontrollable
nervous twitch. He was carrying an enormous ledger whose covers bulged, not to say wriggled as if
the contents were in continuous motion.

"I believe I have done it," he cried triumphantly, twitching so violently that he almost dropped
the book. "I have gotten the accounts to balance! Apart from the residual quantum fluctuations, of
course," he added, less enthusiastically.
"Very good," answered the Manager absently. "Now I should like you to take this little girl,
Alice, here and explain to her about quantum uncertainty and fluctuations in the energy of a system and
all that sort of thing." With a wave of farewell to Alice, the Manager turned back to her desk and
began doing something particularly complicated with all the buttons on it. The Accountant led Alice
out quickly before anything further could happen.
They came to a much smaller, more cluttered office which contained a tall, old-fashioned desk
covered in ledgers and with scraps of paper piled all over the floor. Alice looked at one of the open
ledgers. The page was covered with columns of figures, much like other accounts ledgers she had
seen, except that in this one the figures were continually changing slightly as she looked at them.
"Right!" said the rather Victorian figure in front of Alice. "You want to know about Uncertainty
do you, young lady?"

"Yes please, if it is not too much trouble," replied Alice politely.
"Well now," he began, seating himself at his desk. He steepled his fingers together in the
traditional magisterial manner to increase the dignity of his appearance, but this was not a good idea
as just then he gave such a particularly violent jerk that he got his fingers all tangled up, and he had to
stop to unravel them.
"Well now," he repeated, thrusting his hands deep into his pockets for safety. "The thing you
must remember about energy is that it is conserved, which is to say that there is always the same
amount of it. It may convert from one form to another but the total amount is always the same. At least
it is if you take the long view," he added wistfully and sighed, staring mournfully into the distance.
"Isn't it true in the short term then?" asked Alice, who felt that she should say something to keep
the conversation going.

"Well no, not entirely. In fact, not at all, if the term is short enough. You saw the Heisenberg
relation on the notice outside in the Bank didn't you?"
"Oh yes. I was told it gave the terms for the energy loans."
"Well, so it does, in a way, but where do you think the energy for the loans comes from?"
"Why, from the Bank of course."
"Dear me, no!" said the Accountant, looking slightly horrified. "Most certainly not! It would be a
fine thing if the Bank started lending out energy from its own stock!
"No," he went on conspiratorially, looking around him carefully, "It is not widely known, but the
energy does not come from the Bank. In fact it does not really come from anywhere. It is a quantum
fluctuation. The amount of energy that any given system has is not absolutely definite, but will vary up
and down, and the shorter the time over which you measure it the more it is likely to vary.
"In this respect energy is not really at all like money. Money is well conserved in the short term.
If you want to have money for some purpose, you have to get it from somewhere, don't you? You may
take it out of a bank account, or borrow it from someone, or you might even steal it!"
"I wouldn't do that!" cried Alice indignantly, but the Accountant continued, ignoring her.
"No matter where you get it, it has to come from somewhere. If you get more, then someone else
has less. That is what happens in the immediate short term at any rate.
"In the long term it is different; you may get inflation and find there is more and more money
about. Everyone has more, but it does not seem to buy as much as it did. Energy is quite the reverse in
a way. In the long term it is conserved, the total amount stays the same, and you get nothing like
economic inflation. Every year you will need the same amount of energy on average to transfer from
one state in an atom to another. In the short term, though, energy is not well conserved. A particle can
pick up the energy it needs for some purpose without it having to come from anywhere else; it just
appears as a quantum fluctuation. These fluctuations are a consequence of the uncertainly relation:
The amount of energy you have is uncertain, and the shorter the time you have it the more uncertain
the amount you have."
"That sounds terribly confusing," said Alice.
"You do not have to tell me!" answered her companion emphatically. "It is! How would you like
to be an accountant when the figures you are trying to balance are fluctuating all the time?"
"That sounds terrible," cried Alice sympathetically. "How do you manage?"
"Well, I usually try to take as long as I possibly can to do the accounts. That helps a bit. The
longer the period of time that I spend the smaller the residual fluctuations, you see. Unfortunately
people will get impatient and come to me asking if I am planning to take forever to balance the
accounts. That would be the only way to do it, you see," he went on earnestly. "The longer I take, the

smaller the energy fluctuations, so if I did take forever, why then there would be no fluctuations and
my accounts would balance perfectly," he cried triumphantly. "Unfortunately they just won't let me
alone. Everyone is much too impatient and anxious to be off making transitions from one state to
another all the time."
"That is another thing that I wanted to ask about," remembered Alice. "What are all these states
that I keep hearing of? Would you explain them to me please?"
"I am not really the best person to do that. It is all part of Quantum Mechanics, so you really
ought to go to the Mechanics Institute and ask them there."
"That is what I was told before," said Alice. "If that is the best place to ask, would you please
tell me how I might get there?"
"I am afraid that I cannot actually tell you how to get there. That is not the way we do things
here. But I can arrange that it is very probable that you will get there.
He turned to the far wall of his office, which was covered with a dusty curtain. When he drew
this aside with a sudden jerk, Alice could see a row of doors spaced along the wall. "Where does
each of those lead?" she asked. "Does one of them lead to this Institute you were talking about?"
Each of them could lead you almost anywhere, including, of course, to the Institute. But the point
is that all of them will be very likely to lead you to the door of the Institute."
"I do not understand," complained Alice, with an all-too-familiar feeling of increasing
confusion. "What is the difference? If each of them can lead almost anywhere, it is the same as saying

that they all could lead almost anywhere."
"Not at all! It is a different thing entirely. If you were to go through any one door, why then you
might end up almost anywhere, but if you go through them all at once then you will most probably end
up where you want to be, at the peak of the interference pattern."
"What nonsense!" cried Alice. "I cannot possibly go through all the doors at once. You can only
go through one door at a time you see."
"Ah, that is different! Of course, if I see you going through a door, then you will go through that
door and no other, but if I do not see you, then it is quite possible for you to have gone through any
door. In that case the general rule will apply."
With a wave of his hand he indicated a large, striking notice which was fixed on the wall in front
of his desk, where he could not avoid seeing it. It read:
"That is one of the most basic rules that we have here. If it is possible to do several things, you
do not just do one of them, you have to do them all. That way it saves having to make your mind up
very often. So off you go, just go out through all the doors and when you have, then set off in every
direction at once. You will find it is quite easy and very soon you will have got to the right place."
"This is ridiculous!" protested Alice. " There is no way that I can go through several doors at
once!"
"How can you say that until you have tried? Have you never done two things at the same time?"
"Well, of course I have" answered Alice. "I have watched television while I was doing my
homework, but that is not the same thing at all. I have never gone in two directions at the same time."
"I suggest then that you try it," replied the Accountant, rather huffily. "You never know whether
you can do something until you try. That is the sort of negative thinking which is always holding back
progress. If you want to get anywhere here you have to do everything that you possibly can and do it
all at the same time. You do not have to worry about where it will take you, the interference will take
care of that!"
"How do you mean? What is interference?" cried Alice.
"No time to explain. The Mechanics will tell you all about that. Now off you go and they will
explain when you arrive."
"This is really too bad!" thought Alice to herself. "Everyone I speak to rushes me on somewhere
else and promises me that I will get an explanation as soon as I get there. I wish that someone would

just explain things properly, once and for all! I am sure that I do not know how I can possible go
several ways at the same time. It seems to me to be quite impossible, but he is so certain that I shall
be able to manage it here that I had better try, I suppose."
Alice opened a door and stepped through.
Alice's Many Paths
Alice stepped through the left-hand door and found herself in a small cobbled square with three
narrow alleys leading out of it. She walked down the left-hand alley. Before she had gone very far,
she found herself on the edge of a broad paved area. In the center rose a tall dark building with no
windows on the lower levels. It looked very forbidding.
narrow alleys leading out of it. She walked down the right-hand alley. Before she had gone very far
she came to a park, with weed-choked gravel paths winding between dismal drooping trees. Tall iron
railings surrounded the park and a dank mist obscured the scenery within.
narrow alleys leading out of it. She walked down the middle alley. Before she had gone very far she
came to another small square, in front of a rather shabby-looking building.
Alice stepped through the right-hand door and found herself in a narrow alleyway with two
others branching off it. She walked down the left-hand alley. Before she had gone very far she found
herself on the edge of a broad paved area. In the center rose a tall, dark building with no windows on
the lower levels. It looked very forbidding, and she had a distinct feeling that she ought not to be
there.
others branching off it. She walked down the right-hand alley. Before she had gone very far she came
to a park, with weed-choked gravel paths winding between dismal drooping trees. Tall iron railings
surrounded the park and a dank mist obscured the scenery within. She had a very strong feeling that
she ought not to be there.

others branching off it. She walked on down the central alley. Before she had gone very far she came
to another small square, in front of a rather shabby-looking building. Somehow it seemed to her that
this was the right place to be.
Alice stepped through the center door and found herself facing a wall with three arched
gateways which led to alleys beyond. She walked down the left-hand alley. Before she had gone very
far she found herself on the edge of a broad paved area. In the center rose a tall, dark building with no
windows on the lower levels. It looked very forbidding. She now felt very strongly that she ought not
to be there.
gateways which led to alleys beyond. She did not walk down the right-hand alley at all, as that route
somehow seemed to be completely wrong.
gateways which led to alleys beyond. She walked through the gateway to the central alley. Before she
had gone very far she came to another small square, in front of a rather shabby-looking building. She
now felt quite sure that this was the place where she ought to be.
Alice looked more closely at the building. On a faded board by the door she could make out the
words "Mechanics Institute." This was indeed where she had intended to come!

lice examined the building in front of her. It was unremarkable, a plain brick structure
now rather the worse for wear. In front of her was the board which stated that this was "The
Mechanics Institute." Beside this was a wooden door on which someone had pinned a note: "Don't
knock. Just come in." Alice tried the door and found it was not locked, so she opened it and walked
through.
Inside she found herself in a large, dark room. In the middle of the room there was an area of
light and clarity. Within this limited area it was possible to make out a reasonable amount of detail.
Beyond this there was a seemingly limitless expanse of darkness within which nothing meaningful
could be discerned. In the pool of light was a billiards table, with two figures moving around it.
Alice walked toward them, and as she approached they turned to look at her. They were an oddly
assorted couple. One was tall and angular. He wore a starched white shirt with a tall stiff collar, a
narrow tie, and, rather to Alice's surprise, a boiler suit. His face was aquiline, with bushy side
whiskers. He regarded her with a gaze of such piercing intensity that Alice felt he could clearly
distinguish every tiniest detail in whatever he saw. His companion was smaller and younger. He had
a round face decorated with large, round metal-rimmed glasses. Behind the glasses his eyes were
strangely hard to see; it was difficult to say where he was looking, or even exactly where his eyes
were. He was wearing a white laboratory coat, which was open to display beneath it a T-shirt with a
picture of something vaguely atomic on the front. It was not easy to say exactly what it was meant to
be as the colors appeared to have run in the wash.
"Excuse me, is this the Mechanics Institute please?" asked Alice, mostly for the sake of making
conversation. She knew from the notice outside that it must be.
"Yes, my dear girl," said the taller and more impressive looking of the two. "I myself am a
Classical Mechanic from ClassicWorld, and I am visiting my colleague here, who is a Quantum
Mechanic. Whatever your problem is, I am sure that between us we will be able to assist you, if you
would just wait a moment while we finish our shots."
Both men turned back to the billiards table. The Classical Mechanic took careful aim, clearly
judging all the angles involved to within a tiny fraction of a degree. At last, he very deliberately
played his shot. The ball bounced to and fro in a remarkable series of ricochets, ending in a collision
with the red ball and knocking it squarely into the center of a hole. "There you are," he exclaimed
with satisfaction as he retrieved the ball from the pocket. "That is the way to do it, you know; careful
and exact observation followed by precise action. If you do things that way you can produce any
result you choose."

His companion did not respond, but took his place at the table and made a vague stab with his
cue. After her previous recent experiences Alice was not really surprised to discover that the ball
shot off in every direction at once, so that there was no part of the table where she could say
definitely that the ball had not gone, though equally she could in no way say where it actually was.
After a moment the player went over and peered into one of the pockets, then reached in and drew out
a red ball.
"If you do not mind my saying so," said Alice, "you do seem to play the game very differently."
"Quite so," replied the Classical Mechanic. "I hate the way he plays his shots like that. I like
everything to be done very carefully and precisely and to be planned in every detail in advance.
However," he added, "I imagine that you did not come here to watch us play billiards, so tell us what
you wanted to know."
Alice recounted all her experiences since she came into Quantumland and explained how

confusing she found it and how everything seemed so strange and somehow indefinite. "And I do not
even know how I came to find this building," she finished. "I was told that the interference would
probably bring me to the right place, but I do not understand what happened at all."
"Well now," began the Classical Mechanic, who seemed to have appointed himself as the
spokesman for the two. "I cannot say that I really understand all of it either. As I have said, I like
things to be clear-cut, with cause following effect in a sensible fashion and everything clear and
predictable. If truth be told, not a lot that goes on here makes much sense to me," he whispered
confidentially to her. "I am just visiting from ClassicWorld. That is a splendid place where
everything happens with mechanical precision. Cause follows effect in a wonderfully predictable
fashion, so it all makes sense and you know what is going to happen. What is more, the trains all run
on time," he added as an afterthought.
"That sounds very impressive," said Alice politely. "If it is so well organized, is everything run
by computers?"
"Well, no," answered the Classical Mechanic. " We do not use computers at all. In fact
electronics will not work in ClassicWorld. We are bet ter with steam engines. I do not really feel at
home in Quantumland. My friend here is much more familiar with quantum conditions.
"However," he went on more confidently, "I can tell you what interference is. That happens in
classical mechanics as well. Just follow me and I shall demonstrate how it works."
He led Alice out through a door, down a short corridor, and into another room. This one was
well illuminated, with a clear light which was equally bright everywhere and did not seem to come
from any particular source. They stood on a narrow wooden walkway which ran around the edges of
the room. The floor in the center was covered with some sort of shimmering grayish material, which
did not look solid. It was shot through with random flashes of light, rather like a television set when
there is no picture being received.
Her guide explained, "This is the gedanken room, which means a 'thinking room.' You know that
many gentlemen's clubs have a writing room and a reading room. Well, we have a thinking room. In
here one's thoughts can take on substance, so that anyone can look at them. It allows us to do thought
experiments. These allow us to work out what would happen in various physical cases, and they are
much cheaper than real experiments of course."
"How does it work?" asked Alice. "Do you just think of something and it appears?"
"That is correct; in essence that is all you have to do."
"Oh please, may I try?" asked Alice.
"Yes certainly, if you wish."
Alice thought very intensely at the shifting, flickering surface. To her surprise and delight, where

there had before been a featureless area there was now a group of small furry rabbits hopping about.
"Yes, very pretty," said the Mechanic rather impatiently. "But this is not helping to explain
interference." He made a gesture and all the rabbits vanished, all but one little one who remained
unnoticed in a corner of the area.
"Interference," he began authoritatively, "is something which happens with waves. You can have
all kinds of waves in physical systems, but it will be simplest to consider water waves." He stared
hard at the floor, which turned before Alice's eyes into a sheet of water, with gentle ripples running
over the surface. In the corner the rabbit vanished below the surface with a "plop" as the floor turned
to water beneath it. It quickly struggled out again and glared at them. Then it shook itself, looked
mournfully at its damp fur, and vanished.
"Now we want some waves," continued the Classical Mechanic, paying no attention to the
unhappy rabbit. Alice obligingly thought at the floor and a long curling wave came sweeping across
the surface and broke dramatically upon a beach at one end.
"No, that is not the sort of wave that we want. Those large breaking waves are too complicated.
We want the sort of gentler ripple which spreads out when you throw a stone into water." As he
spoke a series of circular ripples spread out from the middle of the water.
"But we need to think about what are called plane waves where they all move in the same
direction." The circular ripples changed to a series of long, parallel furrows, like a wet plowed field,
all moving across the floor from one side to the other.
"Now we put a barrier in the middle." A low fence sprang up across the center, dividing the
floor in two. The waves flowed up to the barrier and lapped up and down against it, but there was no
way for them to get through and the water beyond was now still and calm.
"Now we make a hole in the barrier, so that the waves can get through there." A neat little gap
appeared just to the left of the fence's center point. Where the ripples struck this narrow gap they
could pass through and spread out in circular ripples into the calm region beyond.
"And now, see what happens when we have two holes in the barrier," cried the Mechanic.
Abruptly there were holes both to the right and to the left of the center. Circular ripples spread out
from both of these. Where they crossed, Alice could see that in some places the water was surging up
and down much more than it had when there was only one hole open, whereas in other places it
hardly moved at all and was locally quite still.
"You can see what is happening if we freeze the motion. We can do that of course in a thought
experiment." All motion on the water stopped, and the patterns of ripples were frozen into position,
as if the whole area had turned abruptly to ice.
"Now we shall mark regions of maximum and minimum amplitude," continued the Classical
Mechanic determinedly. "The amplitude is the amount by which the water moves from the surface
level it had when calm." Two fluorescent arrows appeared, hanging in space above the surface. One

was an apple green color and was pointing down at a point where the disturbance was greatest. The
other was a pale red and pointed to a spot where the surface was almost undisturbed.
"You will be able to see what is happening if we now look at the effect of only one hole at a
time," he said, with steadily increasing enthusiasm. One of the gaps in the fence vanished, and there
were left only the circular ripples spreading out from the other one, though still frozen in position as
if they were made from glass. "Now we will switch to the other hole." Alice could see very little
difference when this happened. The position of the gap had moved and the pattern of circular ripples
coming from it had moved very slightly, but overall it looked much the same. "I am afraid that I cannot
understand what you are trying to show me," she said. "The two cases look just the same to me."
"It will help you to see the difference if we cut quickly from one case to the other." Now the gap
in the fence leapt to and fro, first to the right, then to the left. As it moved, the pattern of ripples on the
surface shifted slightly back and forth.
"Look at the wave patterns under the green arrow," cried the Mechanic, who seemed to Alice to
have become quite unnecessarily excited about the subject. However, she did as requested and saw
that at the point indicated there was a hump in the water in each case. "Each gap in the fence has
produced a wave which is high at this particular point, so when both gaps are open the wave is twice
as high here and the overall rise and fall of the water is much greater than it is for one gap alone. This
is called constructive interference.
"Now look at the wave patterns under the red arrow." Here Alice saw that, while one gap gave
rise to a hump at that point, the other produced a trough in the surface. "You can see that in this
position the wave from one gap goes up and that from the other goes down, so when you have the two
present together, they cancel one another out and you get no overall effect. This is called destructive
interference.
"That is all there is to wave interference really. When two waves overlap and combine with one
another, their amplitudes, the amounts by which they go up or down, combine with one another. In
some places the contributing waves are all going in the same direction, so the disturbances add up
and you get a large effect. At other positions they go in different directions and cancel one another
out."
"Yes, I think that I follow that," said Alice. "So you are saying that the doors in the Bank acted
rather like the gaps in the fence here and gave rise to some sort of large effect in the place where I
needed to go and can - celed one another out in other positions. I do not see how that can apply to my
case though. With your water wave you say that there is more of the wave in one place and less in
another because of this interference, but the wave is spread out over the whole area, while I am
always in just one place at any time."
"Exactly!" cried the Classical Mechanic triumphantly. "That is the problem. As you say, you are
in one place. You are more like a particle than a wave, and particles behave quite differently in a
sensible classical world. A wave is spread over a wide area and you look at only a small portion of
it at any position. Because of interference you may get more or less of it at different positions, but it is
only a small part of the whole wave wherever you look. A particle, on the other hand, is located at

some point. If you look in various positions you will either find the whole particle or it is simply not
there. In classical mechanics there is no question of particles showing interference effects, as we can
show."
He turned to the floor of the gedanken room and stared firmly at it. The surface turned from
water to a smooth area of steel armor, with armored barriers around the edges, high enough for them
to hide behind. Across the middle of the floor, where the low fence had stretched across the water,
was now a tall armored wall, with a narrow slit slightly to the left of center. "Now we can look at the
same setup, but I have changed it so that we can look at fast particles. These are something like
bullets from a gun, so that is what we will use."
He gestured toward one end of the room where there appeared an unpleasant-looking machine
gun with many boxes of ammunition stacked beside it. "This gun has an unsteady mounting, so that it
will not always shoot in the same direction. Some of the bullets will strike the gap in the wall and
pass through, as part of the wave did in our last thought experiment. Most of them, of course, will hit
the steel wall and bounce off. Oh that reminds me," he added abruptly. "We had better wear these in
case we are struck by ricocheting bullets." He produced a pair of steel helmets and handed one to
Alice.
"Do we really need these?" asked Alice. "If this is only a thought experiment, surely these are
thought bullets, and can't do us any harm."
"Well, perhaps so. But you might still think that you had been hit by a bullet, and that would not
be very nice you know."
Alice put the helmet on. She could not feel it on her head and did not think that it would be the
least bit of use, but there did not seem to be much point in arguing any further. The Mechanic stood
upright and gave an imperious wave of his hand, and the gun began firing very noisily. The bullets
shot out in an unsteady stream; most hit the armored screen and whined off in all directions, but a few
got through the slits in the barrier and hit the wall opposite. Alice was intrigued to note that when a
bullet hit this wall, it immediately came to a stop and then rose slowly into the air to hang suspended
in space, directly above the point where it had struck the wall.

"As you can see, whereas the water wave was spread out all over the far wall, a bullet will hit it
in one position only. However, in this experiment there is a greater probability that the bullet will
strike the far wall opposite the slit in the screen than there is that it will bounce off the slit edge and
end up a long way off to the side. If we wait for a little we will see how the probability varies for
different points along the wall." As time passed and the air became full of flying bullets, the number
which were suspended above the wall grew steadily. As she watched, Alice could make out a distinct
trend developing.
"There, you see how the bullets which have passed through the slit are distributed along the
wall," remarked the Mechanic as the gun fell silent. "Most have ended up directly opposite the hole,
and the number falls off steadily on either side. Now see what happens when the slit is offset to the
right." With another wave of his hand the hovering bullets dropped to the ground, and the gun began to
fire again. Though the demonstration was noisy and rather unsettling, as far as Alice could see the end
result was just the same as last time. Frankly, it was disappointing.

"As you can see," said the Mechanic with misplaced confidence, "the distribution is similar to
the previous one, but displaced slightly to the right because the center is now opposite the new
position of the slit." Alice could not see any difference at all, but she was prepared to take his word
for it.
"Now," continued the Mechanic dramatically, "see what happens when both slits are open." As
far as Alice could see it did not make the slightest difference, except that, since two slits were now
open, more bullets got through to hit the far wall. This time she decided to comment. "I am afraid that
it looks just the same to me each time," she said rather apologetically.
"Exactly!" replied the Mechanic with satisfaction. "Except that, as you will of course have
observed, the center of the distribution is now centered between the two slits. We had one
distribution for the probability that bullets will pass through the left-hand slit and another distribution
for the probability that bullets will pass through the right-hand slit. When we have both slits open,
then bullets may pass through either slit, so the overall distribution is given by the sum of the
probabilities that we got for the two slits on their own, since the bullets must have passed through one
or the other. They cannot have passed through both you know," he added, addressing the Quantum
Mechanic, who had just come into the room.
"You say that," replied his colleague, "but how can you be so sure? Just look what happens when
we repeat your gedanken experiment with electrons."

In his turn, the Quantum Mechanic waved his hand at the floor of the room. His gestures were not
so decisive as his companion's, but they seemed to work just as well. The gun and the armored walls
all disappeared. The floor returned to the shimmering material which Alice had first seen, but the
now-familiar wall with two slits near its middle was still there, stretching across the center of the
floor. At the far end of the floor was a wide screen with a greenish glow. "That is a fluorescent
screen," muttered the Mechanic in her ear. "It gives a flash of light when an electron hits it, so it can
be used to detect where they are."
At the opposite end of the floor, where the machine gun had been placed before, was another
gun. This was a small stubby affair, like a very small version of the cannons from which people are
sometimes shot during circus performances. "What is that?" asked Alice.
"Why, it's an electron gun, of course." As Alice looked more carefully, she could see a short
flight of steps leading up to the mouth of the cannon and a line of electrons waiting to be fired from it.
They seemed to be a great deal smaller than when she had last seen them. "But of course," she told
herself, "these are only thought electrons."
As she looked at them, she was surprised to see the electrons all turn and wave to her. "I wonder
how they know me?" she asked herself. "But then I suppose that they are all the same electron that I
met before!"
"Commence firing!" commanded the Quantum Mechanic, and the electrons hurried up the steps
into the gun and shot out in a steady stream. Alice could not make them out at all when they were in
flight, but she saw a bright flash where each one hit the screen. As each flash died, it left a small
glowing star which rose up the screen and remained behind to provide a marker for the position
where the electron had landed.
As had been the case for the machine gun before it, the electron gun continued to fire out its
stream of electrons and the stacks of little glowing stars began to build up a recognizable distribution.
At first Alice could not be too sure what she was seeing, but as the number of little stars displayed
became larger it was clear that their distribution was quite different from that represented by the
previous stacks of bullets.
Instead of a slow, steady decrease from a maximum number in the center, the stars were now
arranged in bands, with dark gaps between where there were few if any of the glowing markers.
Alice realized that this was in a way like the case she had seen for the water waves, where there had
been regions of high activity with calm areas in between. Now there were regions where many
electrons had been detected, with very few in the areas between. It consequently came as no great
surprise to her when Quantum Mechanic said, "There you see a clear interference effect. With the
water waves you had regions of greater and lesser motion at the surface. Now each electron will be
detected at one position only, but the probability of detecting an electron varies from one position to
another. The distribution of different wave intensities which you saw before is replaced by a
probability distribution. With one or two electrons such a distribution is not obvious, but when you
use a lot of electrons you will find more of them in the regions of high probability. With one slit alone
we would have seen that the distribution would decrease smoothly to either side, much as the bullets
or the water waves did when there is only one slit. In this case we see that, when there are two slits

open, the amplitudes from the two slits are interfering and are producing obvious peaks and troughs in
the probability distribution. The behavior of the electrons is quite different from that of my friend's
bullets."
"I do not understand," said Alice. This seemed to her to be the only thing she ever said. "Do you
mean that there are so many electrons going through that somehow the electrons which go through one
hole are interfering with the ones which go through the other?"
"No, that is not what I mean. Not at all. You shall now see what happens when there is only one
electron in flight at any time." He clapped his hands and cried "OK! Let's do it again, but slowly this
time." The electrons sprang into action or rather, to be strictly accurate, one climbed up into the
cannon and shot off. The others continued to sit around where they were. A little later another
electron climbed in and was fired on its way. This continued for some time, and Alice could see the
same pattern of clumps and gaps appearing. These clumps and gaps were not so clear this time as they
had been before because the slow rate at which the electrons were arriving meant that there were not
very many in the clumps, but the pattern was clear enough. "There, you see that the interference effect
works just as well even when there is only one electron present at any time. One electron on its own
can show interference. It can go through both slits and interfere with itself, so to speak."
"But that is silly!" cried Alice. "One electron cannot go through both slits. As the Classical
Mechanic said, it just isn't sensible." She went up to the barrier and peered more closely, to try and
see where the electrons went as they passed through the slits. Unfortunately the light was poor and the
electrons moved by so quickly that she could never quite make out which slit any one had passed
through. "This is ridiculous," thought Alice. "I need more light." She had forgotten that she was in the
"thinking room" and was startled when an intense spotlight mounted on a stand appeared by her
elbow. Quickly she directed the light toward the two slits and was pleased to find that now there was
a visible flash near one hole or the other when the electron passed through. "I have done it!" she
cried. "I can see the electrons as they go through the slits, and it is just as I said it must be. Each one
does go through just one slit."

"Aha!" replied the Quantum Mechanic meaningfully. "But have you looked to see what is
happening to the interference pattern?" Alice looked back toward the far screen and was amazed to
see that now the distribution of little stars fell smoothly from a central maximum, just like the
distribution that she had seen for the classical bullets. It didn't seem fair somehow.
"That is how it always happens; there is nothing that you can do about it," said the Quantum
Mechanic soothingly. "If you don't have any observation to show which hole the electrons go through,
then you get interference between the effects of the two holes. If you do observe the electrons, then
you find that indeed they are in one place or the other, not both, but in that case they also act as you
would expect if they had come through one hole only and you do not get any interference. The
problem is that there is no way in which you can look at the electrons without disturbing them, as
when you shone that light on them, and the very act of making the observation forces the electrons to
choose one course of action. It doesn't matter whether or not you make a note of which hole the
electron came through. It does not matter whether you are aware which hole it came through. Any
observation which could tell you this will disturb the electron and stop the interference. The
interference effects only happen when there is no way that you could know which slit the electron
went through. Whether or not you do know does not matter.

"So you see, when there is interference it seems as if each electron is going through both slits. If
you try and check on this, you will find that the electrons go through only one slit, but then the
interference vanishes. You can't win!"
Alice thought about this for a bit. "That is utterly ridiculous!" she decided.
"Certainly it is," replied the Mechanic with a rather smug smile. "Quite ridiculous I agree, but as
it also happens to be how Nature works we have to go along with it. Complementarity, that's what I
say!"
"Would you please tell me what you mean by complementarity?" asked Alice.
"Why of course. By complementarity I mean that there are certain things you cannot know, not all
at the same time anyhow."
"Complementarity doesn't mean that," protested Alice.
"It does when I use it," replied the Mechanic. "Words mean what I choose. It is a question of
who is to be master, that is all. Complementarity, that's what I say."

"You said that before," pointed out Alice, who was not entirely convinced by his last assertion.
"No, I didn't," said the Mechanic. "This time it means that there are questions you cannot ask of a
particle, such as where it is and, at the same time, how fast it is going. In fact it may not be really
meaningful to talk about an electron having an exact position."
"That is a great deal for one word to mean!" said Alice tartly.
"Why, to be sure," answered the Mechanic, "but when I make a word do extra work like that I
always pay it more. I am afraid that I cannot really explain what is happening to the electrons. An
explanation is usually required to make sense in terms of things you already know about and quantum
physics doesn't do that. It seems to make nonsense but it works. It is probably safe to say that no one
really understands quantum mechanics, so I cannot explain, but I can tell you how we describe what
goes on. Come into the back room and I will do my best."
They left the gedanken room, whose floor had returned to its original shimmering aspect, and
walked down the corridor to another room furnished with scattered armchairs. When they had seated
themselves, the Quantum Mechanic continued. "When we talk about a situation like the electrons
passing through the slits, we describe it with an amplitude. This is something like the waves that you
looked at, and indeed it is often called a wave function instead. The amplitude can pass through both
the slits, and it is not always positive, like a probability. The lowest probability that you can have is
zero, but the amplitude may be negative or positive, so the parts from different paths can cancel or
add and give interference, again just like the water wave."
"So where are the particles?" asked Alice. "Which slit do they actually go through?"
"The amplitude doesn't really tell you about that. However if you square the amplitude, that is

multiply it by itself so that it gives something that is always positive, then it gives you a probability
distribution. If you choose any position this will tell you the probability that, when you observe a
particle, you will find it at that position."
"Is that all it can tell you?" exclaimed Alice. "I must say that it sounds very unsatisfactory. You
would never know where anything is going to be."
"Yes, that is true enough. For one particle you cannot tell where it will be found, except that it
will not be at a position where there is zero probability of course. If you have a large number of
particles, though, then you can be fairly sure that you will find more where the probability is high and
far fewer where it is low. If you have a very large number of particles, then you can say quite
accurately how many will end up where. That was the case with those builders you were telling us
about. They knew what they would get because they used a large number of bricks. For really large
numbers the overall reliability is very good."
"And there is no way you can say what each particle is doing until it is observed?" repeated
Alice, just to get this clear.
"No, no way at all. When the thing that you actually observe could have come about in several
different ways, then you have an amplitude for each possible way, and the overall amplitude is given
by adding all of these together. You have a superposition of states. In some sense the particle is
doing all the things which it could possibly be doing. It is not just that you do not know what the
particle is doing. The interference shows that the different possibilities are all present and affect one
another. In some way they are all equally real. Everything that is not forbidden is compulsory."
"Oh, I saw that on a notice in the Bank. It looked very stern."
"You had better believe it! It is one of the main rules here. Where there are several things which
might happen, they all do. Look at the Cat, for example."
"What cat?," asked Alice, looking around her in confusion.
"Why Schrödinger's Cat over there. He left it with us to look after." Alice looked over in the
corner where the Mechanic was pointing and saw a large tabby cat sleeping in a basket in the corner.
As if awakened by hearing its name the cat stood up and stretched. Or rather, it did and it didn't. Alice
could see that, as well as the slightly hazy figure of the cat standing with back arched in the basket,
there appeared to be another identical cat which was still lying on the bottom. It was very stiff and
motionless and lay in a rather unnatural position. From the look of it, Alice would have sworn that it
was dead.

"Schrödinger devised a gedanken experiment in which an unfortunate cat was enclosed in a box,
together with a flask of poison gas and a mechanism which would break the flask should a sample of
radioactive material happen to decay. Now such a decay is definitely a quantum process. The
material might or might not decay, so according to the rules of quantum physics you would have a
superposition of states, in some of which the decay would have happened and in others it would not.
Of course, for those states where a decay had happened the cat would have been killed, so you would
have a superposition of cat-states, some dead and some alive. When the box was opened someone
would observe the cat, and from that time on it would be either alive or dead. The question which
Schrödinger posed was, 'What was the state of the cat before the box was opened?"'
"And what did happen when the box was opened?" asked Alice.
"Well actually, everyone was so engrossed in discussing the question that no one ever did open
the box, which is why the Cat was left like that."
Alice peered closely into the basket, where one aspect of the Cat was busily licking itself. "He
looks pretty lively to me," she observed. No sooner were the words out of her mouth than the Cat
became fully solid and the dead version vanished. With a satisfied purr the Cat leapt out of the box
and began to stalk a mouse which had just popped out of the wall. Alice noted that there was no
mouse hole visible-the mouse had simply come out of the solid wall. The Quantum Mechanic
followed the direction of her gaze. "Ah, yes. That is an example of barrier penetration; we get it
happening all the time. Where you have a region that a particle could not enter at all according to

classical mechanics, the amplitude does not necessarily stop abruptly at the boundary, though it does
die away rapidly inside the region. If the region is very narrow, then there is still some small
amplitude left at the other side, and this gives a slight probability that the particle may appear there,
having apparently tunneled through an impassible barrier. It happens quite often."
Alice had been thinking through what she had seen and had noted a difficulty. "How is it that I
was able to make an observation and fix the condition of the Cat if it was not able to do it for itself?
What is it that decides when an observation is actually made and who is able to make one?"
"There you have a good question," replied the Quantum Mechanic, "but we are only mechanics
after all, so we do not worry too much about such things. We just get on with the job and use ways
that we know will work in practice. If you want someone to discuss the measurement problem with
you, you will need to go somewhere more academic. I suggest that you go to a class at the
Copenhagen School."
"And how do I get there?" asked Alice, resigned to being passed on somewhere else once again.
In answer the Mechanic led her out into the corridor and opened yet another door. This did not lead
into the alleyway from which she had entered, but into a wood.
Notes
1. Quantum mechanics is usually contrasted with classical or Newtonian mechanics. The latter covers
the detailed description of moving objects which was developed before the early years of the

twentieth century and was based on the original work of Galileo, Newton, and others both before
and since. Newtonian mechanics works very well on a large scale. The motion of the planets can
be predicted over long times and with great accuracy. It works almost as well for artificial
planets and the various exploratory space missions: Their positions may be predicted years
ahead. It also works pretty well for falling apples.
In the case of a falling apple there will be significant resistance from the air that surrounds it.
Classical mechanics describes this as the collision of vast numbers of air molecules bouncing off the
apple. When you ask about air molecules you are told that they are small groups of atoms. When you
ask about atoms there is an embarrassing silence.
Classical mechanics had virtually no success in describing the nature of the world on the scale of
atoms. Things must somehow be different for small objects from how they seem to be for large ones.
If you argue in this way, then you must ask: large or small relative to what? There must be some
dimension, some fundamental constant which fixes the size at which this new behavior becomes
obvious. It is a definite change in the way things are observed to behave, and it is universal. Atoms in
the sun and in distant stars emit light with a spectrum which is like that from a lamp on a table beside
us. The onset of quantum behavior is not something that just happens to take place locally; there is
some fundamental property of Nature involved. This is given by the universal constant ħ, which
features in most equations of quantum mechanics. The world is grainy on the scale defined by this
constant, ħ. On this scale energy and time, position and momentum are blurred together. It need hardly
be pointed out that, on the human scale of perception, ħ is very small indeed and most quantum effects
are not at all obvious.
2. What the Heisenberg uncertainty relations are telling us is that we are looking at things in the
wrong way. We have a preconception that we ought to be able to measure the position and
momentum of a particle at the same time, but we find that we cannot. It is not in the nature of
particles for us to be able to make such a measurement on them, and the theory tells us that we are
asking the wrong questions, questions for which there is no viable answer. Neils Bohr used the
word complementarity to express the fact that there may be concepts which cannot be precisely
defined at the same time: such pairs of concepts as justice and legality, emotion and rationality.
There is, apparently, something fundamentally wrong with our belief that we should be able to talk
about the position and momentum of a particle, or of its exact energy at a given time. It is not clear
why it should be meaningful to talk simultaneously of two such different qualities, but it appears that
it is not.
3. Quantum mechanics is not really about definite particles in the traditional classical sense; instead
you talk about states and amplitudes. If you square an amplitude (i.e., multiply it by itself), then
you get a probability distribution which gives the probability of obtaining various results when
you make an observation or measurement. The actual value that you get for any one measurement
appears to be quite random and unpredictable. So it does look as if the suggestion made earlier
that nature is uncertain and "anything goes" must, after all, be true, does it not?
Well, no-if you make many measurements the average result is accurately predictable. Bookmakers
do not know which horse will win each race, but they confidently expect to make a profit at the end of

the day. They do not anticipate large surprise losses even though they have to work with rather small
numbers, so that the averaging is not too reliable. The number of gamblers will be a mere few
thousand people rather than the 1,000,000,000,000,000,000,000,000 or more atoms you will get in
even a tiny speck of matter. This looks less like a number than a repetitive wallpaper pattern, but it is
undeniably large. The overall statistical fluctuations to be expected for measurements made on such a
large number of atoms are negligible, even though the result for each individual atom may be quite
random.
Quantum-mechanical amplitudes may be calculated very accurately and compared with experiments.
An often quoted result is for the magnetic moment of the electron. Electrons spin like little tops and
they also have electrical properties: They behave rather like tiny bar magnets. The magnetic strength
and the electron spin are related, and their ratio may be calculated using suitable units.
A classical calculation gives the result 1 (with rather arbitrary assumptions about the distribution of
the electric charge in an electron).
The quantum calculation gives the result 2.0023193048 (±8) (the error is in the last decimal place).
A measurement has given the result 2.0023193048 (±4).
This is good agreement! The probability of getting by chance a value which is in such good agreement
is similar to the probability of throwing a dart at random and hitting the bull's-eye on a dartboard-
when the dartboard is as far away as the Moon. This particular result is often given as an example of
the success of quantum theory. It is possible to calculate accurately the amplitudes for other processes
just as accurately, but there are very few quantities which you can measure to this precision.

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